A021029 -id:A021029 - cp's OEIS frontend

A001240 Expansion of 1/((1-2x)(1-3x)(1-6x)).

1, 11, 85, 575, 3661, 22631, 137845, 833375, 5019421, 30174551, 181222405, 1087861775, 6528756781, 39177307271, 235078159765, 1410511939775, 8463200647741, 50779591044791, 304678708005925

Offset: 1

N. J. A. Sloane

Differences of reciprocals of unity.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 228.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..100
Aung Phone Maw, Aung Kyaw, Recursive Harmonic Numbers and Binomial Coefficients, arXiv:1711.10716 [math.CO], 2017.
Mircea Merca, Some experiments with complete and elementary symmetric functions, Periodica Mathematica Hungarica, 69 (2014), 182-189.
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (11,-36,36).

Right-hand column 2 in triangle A008969.
a(n) = A112492(n+1, 3).
Cf. A021029 (partial sums).

A001240:=-1/((6*z-1)*(3*z-1)*(2*z-1)); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

CoefficientList[Series[1/((1-2x)(1-3x)(1-6x)),{x,0,25}],x] (* or *) LinearRecurrence[{11,-36,36},{1,11,85},25] (* Harvey P. Dale, May 15 2011 *)

a(n)=(6^n-2*3^n+2^n)/2 \\ Charles R Greathouse IV, Feb 19 2017

a(n) = 11a(n-1) - 36a(n-2) + 36a(n-3). - John W. Layman
a(n) = (6^n - 2*3^n + 2^n)/2. Also -x^2/6*Beta(x, 4) = Sum_{n>=0} a(n)*(-x/6)^n. Thus x^2*Beta(x, 4) = x - 11/6*x^2 + 85/36*x^3 - 575/216*x^4 + 3661/1296*x^5 - ... . - Vladeta Jovovic, Aug 09 2002
a(n) = Sum_{0<=i,j,k,<=n, i+j+k=n} 2^i*3^j*6^k. - Hieronymus Fischer, Jun 25 2007
a(n) = 2^n + 3^(n+1)*(2^n-1). - Hieronymus Fischer, Jun 25 2007
a(n) = Sum_{k = 0..n-1} 2^(n-2-k) * (3^n - 3^k). - J. M. Bergot, Feb 05 2018

A195116 a(n) = (2+3^n)*(3+2^n).

Original entry on oeis.org

12, 25, 77, 319, 1577, 8575, 48977, 286759, 1699817, 10137775, 60645377, 363332599, 2178384857, 13065493375, 78378545777, 470228096839, 2821239178697, 16927047127375, 101561119454177, 609363227843479, 3656168902513337, 21936982025631775

Offset: 0

Bruno Berselli, Sep 09 2011

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-47,72,-36).

Cf. A060013 ((1+2^n)*(2+1) with n>3).

Cf. A021029 (for the recurrence).

Magma
```
[(2+3^n)*(3+2^n): n in [0..21]];
```

Table[(2 + 3^n) (3 + 2^n), {n, 0, 30}] (* Vincenzo Librandi, Mar 26 2013 *)

for(n=0, 21, print1((2+3^n)*(3+2^n)", "));

def a(n): return (2+3**n)*(3+2**n)
print([a(n) for n in range(23)]) # Michael S. Branicky, Dec 25 2021

G.f.: (12-119*x+341*x^2-294*x^3)/((1-x)*(1-2*x)*(1-3*x)*(1-6*x)).

Sum_{i=0..n} a(i) = (1/10)*(12*6^n+45*3^n+40*2^n+60*n+23).

cp's OEIS Frontend

A001240 Expansion of 1/((1-2x)(1-3x)(1-6x)).

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